Extensions of theorems of Gasch\"utz, \v{Z}mud$'$ and Rhodes on faithful representations

TL;DR

This paper generalizes classical theorems on faithful representations of finite groups to finite semigroups over arbitrary fields, determining the minimal number of irreducible constituents needed for such representations.

Contribution

It extends Gasch"utz, mud, and Rhodes' theorems to finite semigroups, providing a unified framework for understanding faithful completely reducible representations over any field.

Findings

Determined the minimum number of irreducible constituents in faithful semigroup representations.

Extended classical group representation theorems to semigroups.

Developed a relativized version of mud's theorem for groups.

Abstract

Gasch\"utz (1954) proved that a finite group $G$ has a faithful irreducible complex representation if and only if its socle is generated by a single element as a normal subgroup; this result extends to arbitrary fields of characteristic $p$ so long as $G$ has no nontrivial normal $p$-subgroup. \v{Z}mud$'$ (1956) showed that the minimum number of irreducible constituents in a faithful complex representation of $G$ coincides with the minimum number of generators of its socle as a normal subgroup; this result can also be extended to arbitrary fields of any characteristic $p$ such that $G$ has no nontrivial normal $p$-subgroup (i.e., over which $G$ admits a faithful completely reducible representation). Rhodes (1969) characterized the finite semigroups admitting a faithful irreducible representation over an arbitrary field as generalized group mapping semigroups over a group admitting a faithful irreducible representation over the field in question. Here, we provide a common generalization of the theorems of \v{Z}mud$'$ and Rhodes by determining the minimum number of irreducible constituents in a faithful completely reducible representation of a finite semigroup over an arbitrary field (provided that it has one). Our key tool for the semigroup result is a relativized version of \v{Z}mud$'$'s theorem that determines, given a finite group $G$ and a normal subgroup $N\lhd G$, what is the minimum number of irreducible constituents in a completely reducible representation of $G$ whose restriction to $N$ is faithful.